Question

The solution of ${\int }_{\sqrt{2}}^x\frac{dt}{t\sqrt{t^2-1}}=\frac{\pi }{12}$ is

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Answer 1

Answer: (B)

Given,
${\int }_{\sqrt{2}}^x\frac{dt}{t\sqrt{t^2-1}}=\frac{\pi }{12}$
$\Rightarrow {\left[{\sec }^{-1}t\right]}_{\sqrt{2}}^x=\frac{\pi }{12}$
$\Rightarrow {\sec }^{-1}x-{\sec }^{-1}(\sqrt{2})=\frac{\pi }{12}$
$\Rightarrow {\sec }^{-1}\times \frac{\pi }{4}=\frac{\pi }{12}$
$\Rightarrow {\sec }^{-1}x=\frac{\pi }{12}+\frac{\pi }{4}=\frac{4\pi }{12}=\frac{\pi }{3}$
$\therefore x=\sec \frac{\pi }{3}=2$